Abstract

We apply the Krasnoselskii fixed-point theorem to investigate the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with a parameter; some verifiable sufficient results are established easily. In particular, our results extend and improve some previous results.

1. Introduction

It is well known that impulsive differential equations arise naturally from a wide variety of applications such as aircraft control, the inspection processes in operations research, drug administration, and threshold theory in biology. Therefore, the impulsive differential equations represent a more natural framework for the mathematical model of many real world phenomena than differential equations (see [1–7]). In recent years, many researchers have obtained some properties of impulsive differential equations, such as oscillation, asymptotic behavior, stability and existence of solutions (see [8–16]). However, there are a little work discussing the existence of multiple positive periodic solutions for the high-dimensional functional differential equations with impulse and parameters. Motivated by this, in this paper, we mainly consider the following impulsive functional differential equations with a parameter:
where is a parameter, , , are -periodic. , is an operator on (here denoting the Banach space of bounded continuous operator with the norm , where ); . If , then for any , where is defined by for and (here representing the right limit of at the point ). Consider that ; that is, changes decreasingly suddenly at times . is a constant, , , , and . We assume that there exists an integer such that , , where .

It is well known that the functional differential system (1) includes many mathematical ecological models for example:in the case (a), , in [17], Zeng et al. studied the existence of multiple positive periodic solutions of (1) by applying the Krasnoselskii fixed-point theorem.in the case (b), , ; in [18], Zhang et al. established the existence of positive periodic solutions of (1) by using the fixed-point theorem in cones.in the case (c), , , and ; in [19], Jiang et al. investigated the existence, multiplicity, and nonexistence of positive periodic solutions of (1).In this paper, we will study the existence of positive periodic solutions in more cases than the previously mentioned papers and obtain some easily verifiable sufficient criteria.

Throughout the paper, we make the following assumptions: satisfy Caratheodory conditions; that is, , are locally Lebesgue measurable in for each fixed , are continuous in for each fixed , and are -periodic functions in . Moreover, there exist -periodic functions , , , which are locally bounded Lebesgue measurable such that , and , ; for all , and is a continuous function of for each , ;for any and , there exists such that for , , , and imply that , ;, satisfies and ; , , satisfy Caratheodory conditions and are -periodic functions in and, moreover, for all . There exists a positive constant such that , , . Without loss of generality, we can assume that and .In addition, the parameters in this paper are assumed to be not identically equal to zero.

Furthermore, we will use the following notation. Let denote by the set of operators which are continuous for , and have discontinuities of the first kind at the points but are continuous from the left at these points. For each , the norm of is defined as . The matrix means that each pair of corresponding elements of and satisfies the inequality “” (“”). In particular, is called a positive matrix if .

The paper is organized as follows. In Section 2, we give some definitions and lemmas to prove the main results of this paper. In Section 3, existence theorems for one or two positive periodic solutions of (1) are established by using the Krasnoselskii fixed-point theorem under some conditions.

2. Preliminaries

In this section, we make some preparations for the following sections. For , , we define
It is clear that , , . For all and by , we have
In view of , we also define for the following:
Let with the norm , . It is easy to verify that is a Banach space. Define as a cone in by
We easily verify that is a cone in . We define an operator as follows:
where
The proofs of the main results in this paper are based on an application of the Krasnoselskii fixed-point theorem in cones. To make use of the fixed-point theorem in cones, firstly, we need to introduce some definitions and lemmas.

Definition 1 (see [20]). A function is said to be a positive solution of (1), if the following conditions are satisfied:(a) is absolutely continuous on each ;(b)for each and exist, and ;(c) satisfies the first equation of (1) for almost everywhere in and satisfies the second equation of (1) at impulsive point .

Definition 2 (see [21]). Let be a real Banach space; is a cone of . The semiorder induced by the cone is denoted by “”; that is, if and only if for any .Secondly, let us introduce the Krasnoselskii point theorem in cones which will be used in this paper.

Lemma 3 (for the Krasnoselskii fixed-point theorem; see [22–24]). Let be a cone in a real Banach space . Assume that and are open subsets of with , where . Let be a completely continuous operator and satisfy either(1), for any and , for any , or(2), for any and , for any .Then has a fixed point in .

Lemma 4 (see [25]). Assume that and are continuous nonnegative functions defined on the interval ; then there exists such that

Lemma 5. Assume that hold. The existence of positive -periodic solution of (1) is equivalent to that of nonzero fixed point of in .

Proof. Assume that is a periodic solution of (1). Then, we have
Integrating the above equation over , we can have
where , , , and . Therefore,
which can be transformed into
Thus, is a periodic solution for (7).If and with , then, for any we can get the derivation of (7) about ,
For any , , we have from (7) that
Hence is a positive -periodic solution of (1). Thus we complete the proof of Lemma 5.

Lemma 6. Assume that hold. Then is well defined.

Proof. From (7), it is easy to verify that is continuous in , and exist, and for each . Moreover, for any ,
Therefore, . From (7), we have
Noticing that , we obtain
Therefore, . This completes the proof of Lemma 6.

Lemma 7. Assume that hold. Then is completely continuous.

Proof. We first show that is continuous. By -, and are continuous in ; it follows that, for any , let be small enough to satisfy that, if , with ,
Therefore,
which implies that is continuous on .Next we show that maps a bounded set into a bounded set. Indeed, let be a bounded set. For any and , by (7), we have
Since is bounded, in view of the continuity of , it follows from (19) that is bounded and is uniformly bounded. Finally, we show that the family of functions is equicontinuous on . Let with . From (7), for any , we have
Since for , , , , , and are uniformly bounded in ; in view of (21), it is easy to see that when tends to zero, tends uniformly to zero in . Hence, is a family of uniformly bounded and equicontinuous functions on . By Ascoli-Arzelà theorem, the operator is completely continuous. The proof of Lemma 7 is complete.

For convenience in the following discussion, we introduce the following notations:
where denotes either or , denotes a positive number, and .

3. Main Results

Our main results of this paper are as follows.

Theorem 8. Assume that and the following conditions:, ;hold. Then (1) has two positive -periodic solutions.

Proof. First, we define ; then is an open subset of . From (7), , and Lemma 4, for any , we have
This yields
On the other hand, if holds, then we can choose , such that and for , , and , where constant satisfies . By (7) and Lemma 4, we can obtain
This yields
In view of (24) and (26), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1).Likewise, if holds, then there is such that and for , , and , where constant satisfies . Let and it follows that for , , and . Thus
By (7) and Lemma 4, we have
this yields
In view of (24) and (29), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). Therefore (1) has at least two positive periodic solutions; that is, . This proves Theorem 8.

Remark 9. Assume that and the following conditions:;hold. Then (1) has a positive -periodic solution.

Corollary 10. Assume that and hold.is satisfied; then (1) has two positive -periodic solutions;is satisfied; then (1) has a positive -periodic solution.

Theorem 11. Assume that and the following conditions:;hold. Then (1) has two positive -periodic solutions.

Proof. We define , for a positive number . Then is an open subset of and . By (7), , and Lemma 4, for any , we have
This implies that for any
On the one hand, since , there exists and small enough satisfies such that, for any with ,
Define ; then is an open subset of . For any , by (7) and Lemma 4, we have
This yields
In view of (31) and (34), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). On the other hand, if , we can find small enough that satisfies and large enough , such that ,
Define and ; then is an open subset of . For any , from (7) and Lemma 3, we have
This yields
In view of (31) and (37), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). Therefore, (1) has at least two positive periodic solutions; that is, . This proves Theorem 11.

Remark 12. Assume that and the following conditions:;, or hold. Then (1) has a positive -periodic solution.

Corollary 13. Assume that and hold.is satisfied; then (1) has two positive -periodic solutions;is satisfied; then (1) has a positive -periodic solution.

Theorem 14. Assume that and;hold. Then (1) has a positive -periodic solution, where , , , and are positive constants.

Proof. From , we can choose such that . Thus there exists such that, for , and ,
by (7), , and Lemma 4, we have
This implies that for any
On the other hand, choose such that and , and from , we can obtain
It is easy to see that there exists large enough , such that ,
Define and ; then is an open subset of . From (7), , and Lemma 4, for any , we have
This yields
In view of (40) and (44), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). This proves Theorem 14.

Corollary 15. Assume that and the following condition:hold. Then (1) has a positive -periodic solution.

Similarly, we can prove the following theorem and corollary.

Theorem 16. Assume that and the following conditions:;hold. Then (1) has a positive -periodic solution, where , , , and are positive constants.

Corollary 17. Assume that and the following condition:hold. Then (1) has a positive -periodic solution.

Theorem 18. Assume that , and the following condition:, and , ,hold. Then (1) has two positive -periodic solutions.

Proof. First, we define ; then is an open subset of . From (7), , and Lemma 4, for any , we have
This yields
On the one hand, since and , there exists such that for
Set ; then is an open subset of . From (7), , and Lemma 4, for any , , and , we have
This yields
In view of (46) and (49), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1).On the other hand, if , and hold, then there is such that
for , , . Let and it follows that for , , and . Thus
By (7), , and Lemma 4, we have